Lie algebra presented by generators and relations

Let $V$ be a complex vector space and let $R\subseteq L(V)$ be a subset of the free Lie algebra (Free Lie algebra on a vector space). The Lie ideal generated by $R$ is the smallest Lie ideal $I(R)⊴L(V)$ containing $R$, namely the intersection of all ideals containing $R$; it exists because $L(V)$ itself is such an ideal. The Lie algebra presented by the generators $V$ and the relations $R$ is the quotient Lie algebra $L(V)\mathrm{/}I(R)$ (Quotient Lie algebras). The bracket on the quotient is well defined by The quotient Lie-algebra bracket is well-defined. The images of the elements of $V$ generate $L(V)\mathrm{/}I(R)$ as a Lie algebra, and a Lie algebra homomorphism out of $L(V)\mathrm{/}I(R)$ corresponds to a Lie algebra homomorphism $L(V)\to \mathfrak{g}$ that annihilates $R$, by the universal property of the free Lie algebra (Universal property of the free Lie algebra) together with the universal property of the quotient. In particular, to define a homomorphism from a presented Lie algebra it suffices to prescribe the images of the generators and to verify that all relations are satisfied.